Journal of Topology最新文献

We develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding well-behaved theories of power operations for $�E_{\infty }$ ring spectra. In particular, we consider Quillen cohomology in the context of algebras over algebraic theories, plethories, and Koszul resolutions for algebras over additive theories. By combining this general algebra with obstruction-theoretic machinery, we obtain tools for computing with $�E_{\infty }$ algebras over $�F_p$ and over Lubin–Tate spectra. As an application, we demonstrate the existence of $�E_{\infty }$ periodic complex orientations at heights $��h�⩽�2�$.

We prove that for all $��k�,�m�,�n�\in �N�\cup �\left\{�\infty �\right\}�$ with $��4�⩽�k�⩽�m�⩽�n�$, there exists a finitely generated group $�G$ with a finitely generated subgroup $�H$ such that $��asdim�\left(�G�\right)�=�k�$, $��{asdim}_{�A�N�}��\left(�G�\right)��=�m�$, and $��{asdim}_{�A�N�}��\left(�H�\right)��=�n�$. This simultaneously answers two open questions in asymptotic dimension theory.

We consider the classical Steenrod problem on realization of integral homology classes by continuous images of smooth oriented manifolds. Let $��k�\left(�n�\right)�$ be the smallest positive integer such that any integral $�n$-dimensional homology class becomes realizable in the sense of Steenrod after multiplication by $��k�\left(�n�\right)�$. The best known upper bound for $��k�\left(�n�\right)�$ was obtained independently by Brumfiel and Buchstaber in 1969. All known lower bounds for $��k�\left(�n�\right)�$ were very far from this upper bound. The main result of this paper is a new lower bound for $��k�\left(�n�\right)�$ that is asymptotically equivalent to the Brumfiel–Buchstaber upper bound (in the logarithmic scale). For , we prove that our lower bound is exact. Also, we obtain analogous results for the case of realization of integral homology classes by continuous images of smooth stably complex manifolds.

In this paper, generalizing our previous construction, we equip the relative moduli stack of complexes over a Calabi–Yau fibration (possibly with singular fibers) with a shifted Poisson structure. Applying this construction to the anticanonical linear systems on surfaces, we get examples of compatible Poisson brackets on projective spaces extending Feigin–Odesskii Poisson brackets. Computing explicitly the corresponding compatible brackets coming from Hirzebruch surfaces, we recover the brackets defined by Odesskii–Wolf.

We introduce a new method for proving twisted homological stability, and use it to prove such results for symmetric groups and general linear groups. In addition to sometimes slightly improving the stable range given by the traditional method (due to Dwyer), it is easier to adapt to nonstandard situations. As an illustration of this, we generalize to $�{GL}_n$ of many rings $�R$ a theorem of Borel that says that passing from $�{GL}_n$ of a number ring to a finite-index subgroup does not change the rational cohomology. Charney proved this generalization for trivial coefficients, and we extend it to twisted coefficients.

We introduce the notion of hyperbolic orientation of a motivic ring spectrum, which generalises the various existing notions of orientation (by the groups $�GL$, $�{SL}^c$, $�SL$, $�Sp$). We show that hyperbolic orientations of $�\eta$-periodic ring spectra correspond to theories of Pontryagin classes, much in the same way that $�GL$-orientations of arbitrary ring spectra correspond to theories of Chern classes. We prove that $�\eta$-periodic hyperbolically oriented cohomology theories do not admit further characteristic classes for vector bundles, by computing the cohomology of the étale classifying space $�{BGL}_n$. Finally, we construct the universal hyperbolically oriented $�\eta$-periodic commutative motivic ring spectrum, an analogue of Voevodsky's cobordism spectrum $�MGL$.

We establish motivic versions of the theorems of Lin and Gunawardena, thereby confirming the motivic Segal conjecture for the algebraic group $�{\mu }_{\ell }$ of $�\ell$th roots of unity, where $�\ell$ is any prime. To achieve this we develop motivic Singer constructions associated to the symmetric group $�S_{\ell }$ and to $�{\mu }_{\ell }$, and introduce a delayed limit Adams spectral sequence.

We study the homotopy of the connected sum of a manifold with a projective space, viewed as a typical way to stabilize manifolds. In particular, we show a loop homotopy decomposition of a manifold after stabilization by a projective space, and provide concrete examples. To do this, we trace the effect in homotopy theory of surgery on certain product manifolds by showing a loop homotopy decomposition after localization away from the order of the image of the classical $�J$-homomorphism.